Problem 30 - University Physics with Modern Physics

Question

On-Demand Water Heaters. Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawbacks are that energy is wasted because the tank loses heat when it is not in use and that you can run out of hot water if you use too much. Some utility companies are encouraging the use of on-demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is 2.5 gal/min (9.46 L/min) with the tap water being heated from \(50^{\circ} \mathrm{F}\left(10^{\circ} \mathrm{C}\right)\) to \(120^{\circ} \mathrm{F}\left(49^{\circ} \mathrm{C}\right)\) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

Step-by-Step Solution

Answer

About 25.7 kW of power is needed to heat the water to the desired temperature.

Step 1: Understand the Problem

We need to find the rate of heat input required to raise the temperature of water in a shower from 50°F (10°C) to 120°F (49°C) with a flow rate of 2.5 gallons per minute.

Step 2: Convert Units

Convert the flow rate from gallons per minute to liters per minute since specific heat is typically given in metric units. We have 2.5 gal/min equivalent to 9.46 L/min as given. Convert the temperature change from Fahrenheit to Celsius: \( \Delta T = 49°C - 10°C = 39°C \).

Step 3: Identify the Required Formula

Use the formula for heat transfer: \( Q = mc\Delta T \), where \( Q \) is the heat required, \( m \) is the mass flow rate, \( c \) is the specific heat capacity of water, and \( \Delta T \) is the temperature change.

Step 4: Calculate the Mass Flow Rate

Compute the mass flow rate by converting the volume flow rate from liters per minute to kilograms per minute, knowing that 1 liter of water weighs approximately 1 kg. Therefore, \( 9.46 \text{ L/min} = 9.46 \text{ kg/min} \).

Step 5: Apply Specific Heat Capacity

The specific heat capacity of water is approximately \( 4.186 \text{ J/g°C} \) or \( 4.186 \times 10^3 \text{ J/kg°C} \).

Step 6: Calculate Heat Input per Minute

Substitute the values into the heat transfer formula. \( Q = 9.46 \times 4.186 \times 10^3 \times 39 \). Calculate \( Q \) to find the rate of heat input per minute.

Step 7: Convert to Kilowatts

Since the result is in joules per minute, convert it to watts (1 watt = 1 joule/second) by dividing by 60,000 (as there are 60 seconds in a minute and 1000 watts in a kilowatt): \( \text{kW} = \frac{Q}{60000} \).

Key Concepts

On-Demand Water Heaters

On-demand water heaters, often referred to as flash heaters, are revolutionizing the way we use hot water in our homes. Unlike traditional water heaters that maintain a large tank of water at a continuous temperature, on-demand heaters work only when hot water is needed. This means:

• Energy Efficiency: Since water is heated only when required, there is a significant reduction in energy wastage associated with heat loss from stored water.
• Unlimited Hot Water: You will no longer run out of hot water as these systems heat water continuously, meeting ongoing demand effortlessly.

Additionally, because there is no need for a storage tank, these units are more compact and can easily fit in small spaces. Many utility companies promote their use to enhance overall energy efficiency in households.

Specific Heat Capacity

Specific heat capacity is a critical factor in understanding how much energy is needed to change the temperature of a substance. For water, the specific heat capacity is about 4.186 J/g°C or 4.186 x 10³ J/kg°C. This means that:

• A higher specific heat capacity requires more energy to increase the temperature of a given mass.
• For practical purposes, this value tells us how much energy is needed to raise the temperature of a certain mass of water by one degree Celsius.

In our on-demand heater exercise, knowing the specific heat capacity allows us to compute the energy requirements to heat water efficiently as it flows through the system from the tap inlet to the desired temperature.

Energy Efficiency

Energy efficiency in water heating systems is crucial for reducing overall energy costs and environmental impact. On-demand water heaters are particularly efficient because they eliminate standby heat losses commonly found in conventional tank heaters. Key advantages include:

• Reduced Heat Loss: Heat is only applied when water flows, preventing unnecessary energy consumption.
• Cost Savings: By heating water only as needed, households can see a reduction in their utility bills.

Moreover, their ability to deliver an unending supply of hot water without the need for a large storage system enhances their appeal. As such, the demand for these heaters is increasing in energy-conscious homes.